The Stacks project

20.19 Cohomology and colimits

Let $X$ be a ringed space. Let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system of sheaves of $\mathcal{O}_ X$-modules over the directed set $I$, see Categories, Section 4.21. Since for each $i$ there is a canonical map $\mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ we get a canonical map

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ p(X, \mathcal{F}_ i) \longrightarrow H^ p(X, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

for every $p \geq 0$. Of course there is a similar map for every open $U \subset X$. These maps are in general not isomorphisms, even for $p = 0$. In this section we generalize the results of Sheaves, Lemma 6.29.1. See also Modules, Lemma 17.22.8 (in the special case $\mathcal{G} = \mathcal{O}_ X$).

Lemma 20.19.1. Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties:

  1. there exists a basis of quasi-compact open subsets, and

  2. the intersection of any two quasi-compact opens is quasi-compact.

Then for any directed system $(\mathcal{F}_ i, \varphi _{ii'})$ of sheaves of $\mathcal{O}_ X$-modules and for any quasi-compact open $U \subset X$ the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ q(U, \mathcal{F}_ i) \longrightarrow H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

is an isomorphism for every $q \geq 0$.

Proof. It is important in this proof to argue for all quasi-compact opens $U \subset X$ at the same time. The result is true for $q = 0$ and any quasi-compact open $U \subset X$ by Sheaves, Lemma 6.29.1 (combined with Topology, Lemma 5.27.1). Assume that we have proved the result for all $q \leq q_0$ and let us prove the result for $q = q_0 + 1$.

By our conventions on directed systems the index set $I$ is directed, and any system of $\mathcal{O}_ X$-modules $(\mathcal{F}_ i, \varphi _{ii'})$ over $I$ is directed. By Injectives, Lemma 19.5.1 the category of $\mathcal{O}_ X$-modules has functorial injective embeddings. Thus for any system $(\mathcal{F}_ i, \varphi _{ii'})$ there exists a system $(\mathcal{I}_ i, \varphi _{ii'})$ with each $\mathcal{I}_ i$ an injective $\mathcal{O}_ X$-module and a morphism of systems given by injective $\mathcal{O}_ X$-module maps $\mathcal{F}_ i \to \mathcal{I}_ i$. Denote $\mathcal{Q}_ i$ the cokernel so that we have short exact sequences

\[ 0 \to \mathcal{F}_ i \to \mathcal{I}_ i \to \mathcal{Q}_ i \to 0. \]

We claim that the sequence

\[ 0 \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i \to 0. \]

is also a short exact sequence of $\mathcal{O}_ X$-modules. We may check this on stalks. By Sheaves, Sections 6.28 and 6.29 taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma 10.8.8) we deduce the result. We claim that $H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all quasi-compact open $U \subset X$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i H^{q_0}(U, \mathcal{I}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{q_0}(U, \mathcal{Q}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{q_0 + 1}(U, \mathcal{F}_ i) \ar[d] \ar[r] & 0 \ar[d] \\ H^{q_0}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) \ar[r] & H^{q_0}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i) \ar[r] & H^{q_0 + 1}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \ar[r] & 0 } \]

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_ i$ are injective. The top row is exact by an application of Algebra, Lemma 10.8.8. Hence by the snake lemma we deduce the result for $q = q_0 + 1$.

It remains to show that the claim is true. We will use Lemma 20.11.9. Let $\mathcal{B}$ be the collection of all quasi-compact open subsets of $X$. This is a basis for the topology on $X$ by assumption. Let $\text{Cov}$ be the collection of finite open coverings $\mathcal{U} : U = \bigcup _{j = 1, \ldots , m} U_ j$ with each of $U$, $U_ j$ quasi-compact open in $X$. By the result for $q = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = \mathop{\mathrm{colim}}\nolimits _ i \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_ i) \]

because all the multiple intersections $U_{j_0 \ldots j_ p}$ are quasi-compact. By Lemma 20.11.1 each of the complexes in the colimit of Čech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma 10.8.8 we see that also the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^ p(\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma 20.11.9 are satisfied and the claim follows. $\square$

Lemma 20.19.2. Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system of abelian sheaves on $X$. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$. Let $p \geq 0$ be an integer. Assume the set of opens $V \subset Y$ such that $H^ p(f^{-1}(V), \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(f^{-1}(V), \mathcal{F}_ i)$ is a basis for the topology on $Y$. Then $R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i$.

Proof. Recall that $R^ pf_*\mathcal{F}$ is the sheafification of the presheaf $\mathcal{G}$ sending $V$ to $H^ p(f^{-1}(V), \mathcal{F})$, see Lemma 20.7.3. Similarly, $R^ pf_*\mathcal{F}_ i$ is the sheafification of the presheaf $\mathcal{G}_ i$ sending $V$ to $H^ p(f^{-1}(V), \mathcal{F}_ i)$. Recall that sheafification is the left adjoint to the inclusion from sheaves to presheaves, see Sheaves, Section 6.17. Hence sheafification commutes with colimits, see Categories, Lemma 4.24.5. Hence it suffices to show that the map of presheaves (with colimit in the category of presheaves)

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \longrightarrow \mathcal{G} \]

induces an isomorphism on sheafifications. For this it suffices to show that the presheaves $\mathcal{G}$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ agree on a basis for the topology of $Y$. Namely, in this case the stalks of their sheafifications, which can be computed directly from the presheaf values on elements of the basis, agree. The required agreement is exactly the assumption of the lemma. $\square$

Next we formulate the analogy of Sheaves, Lemma 6.29.4 for cohomology. Let $X$ be a spectral space which is written as a cofiltered limit of spectral spaces $X_ i$ for a diagram with spectral transition morphisms as in Topology, Lemma 5.24.5. Assume given

  1. an abelian sheaf $\mathcal{F}_ i$ on $X_ i$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$,

  2. for $a : j \to i$ an $f_ a$-map $\varphi _ a : \mathcal{F}_ i \to \mathcal{F}_ j$ of abelian sheaves (see Sheaves, Definition 6.21.7)

such that $\varphi _ c = \varphi _ b \circ \varphi _ a$ whenever $c = a \circ b$. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits p_ i^{-1}\mathcal{F}_ i$ on $X$.

Lemma 20.19.3. In the situation discussed above. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and let $U_ i \subset X_ i$ be quasi-compact open. Then

\[ \mathop{\mathrm{colim}}\nolimits _{a : j \to i} H^ p(f_ a^{-1}(U_ i), \mathcal{F}_ j) = H^ p(p_ i^{-1}(U_ i), \mathcal{F}) \]

for all $p \geq 0$. In particular we have $H^ p(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X_ i, \mathcal{F}_ i)$.

Proof. The case $p = 0$ is Sheaves, Lemma 6.29.4.

In this paragraph we show that we can find a map of systems $(\gamma _ i) : (\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ with $\mathcal{G}_ i$ an injective abelian sheaf and $\gamma _ i$ injective. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{I}_ i$ where $\mathcal{I}_ i$ is an injective abelian sheaf on $X_ i$. Then we can consider the family of maps

\[ \gamma _ i : \mathcal{F}_ i \longrightarrow \prod \nolimits _{b : k \to i} f_{b, *}\mathcal{I}_ k = \mathcal{G}_ i \]

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{I}_ k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map

\[ \psi _ a : f_ a^{-1}\mathcal{G}_ i \to \mathcal{G}_ j \]

whose components are the canonical maps $f_ b^{-1}f_{a \circ b, *}\mathcal{I}_ k \to f_{b, *}\mathcal{I}_ k$ for $b : k \to j$. Thus we find an injection $\{ \gamma _ i\} : \{ \mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ of systems of abelian sheaves. Note that $\mathcal{G}_ i$ is an injective sheaf of abelian groups on $X_ i$, see Lemma 20.11.11 and Homology, Lemma 12.27.3. This finishes the construction.

Arguing exactly as in the proof of Lemma 20.19.1 we see that it suffices to prove that $H^ p(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i) = 0$ for $p > 0$.

Set $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i$. To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact open of $X$, it suffices to show that the Čech cohomology of $\mathcal{G}$ for any open covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma 20.11.9. Such a covering is the inverse by $p_ i$ of such a covering $\mathcal{U}_ i$ on the space $X_ i$ for some $i$ by Topology, Lemma 5.24.6. We have

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \check{\mathcal{C}}^\bullet (f_ a^{-1}(\mathcal{U}_ i), \mathcal{G}_ j) \]

by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma 20.11.1. Thus we conclude by Algebra, Lemma 10.8.8. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01FE. Beware of the difference between the letter 'O' and the digit '0'.